Over the past decade, a great deal of scientific attention has been paid to quantitative phase imaging, which has emerged as an important tool for phase visualization and structure retrieval of miniature non-absorbing specimens such as micro-optical elements, unstained cells, and other types of biological and transparent technical samples. Zernike phase contrast [1] and differential interference contrast microscopy [2] have proven to be extremely powerful imaging tools for visualization of phase variation. However, in general the phase information obtained through such means is neither linear nor quantitative, yielding only qualitative descriptions in terms of optical path-length measurement.
Interference techniques such as digital holography microscopy (DHM) are well established methods for quantitative phase measurement [4]. DHM has been successfully demonstrated in the characterization of a microlens array [4], investigations of cellular dynamics [5] and drug-induced morphology changes [6]. However, this class of method typically relies on two-beam interference with a high degree of coherence and thus is usually plagued with problems of phase aberration and coherent noise that prevent accurate phase retrieval and formation of high quality images.
About thirty years ago, Teague [7] derived an equation for wave propagation in terms of phase and intensity distributions, and showed that the phase distribution may be determined by measuring only the intensity distributions. We call this equation the Transport-of-Intensity Equation (TIE).
Let us consider an electro-magnetic wave propagating in a direction z called the optic axis towards an object plane. The two perpendicular directions (transverse spatial coordinates) are denoted by x,y, and the position vector r denotes the position (x,y) in the object plane, i.e. the position as measured with the two transverse spatial coordinates. The wave has an intensity in the object plane denoted by I(r) and a phase in the object plane denoted by φ(r). Originating from the free-space Helmholtz wave equation, the TIE relates the object-plane phase to the first derivative of intensity with respect to the optical axis in the near Fresnel region [7] as follows,
                                                        -              k                        ⁢                                          ∂                                  I                  ⁡                                      (                    r                    )                                                                              ∂                z                                              =                                    ∇              ⊥                        ⁢                          ·                              [                                                      I                    ⁡                                          (                      r                      )                                                        ⁢                                                            ∇                      ⊥                                        ⁢                                          φ                      ⁡                                              (                        r                        )                                                                                            ]                                                    ,                            (        1        )            where k is the wave number 2π/λ, and  is the gradient operator over r.
Suppose that I(r)>0, (note that I(r) can take on the value of zero but to use the TIE, I(r) has to be greater than zero) and with appropriate boundary conditions, the solution to TIE is known to exist and be unique [7]. That is, the phase φ(r) can be uniquely determined by solving TIE using an observed intensity I(r) and longitude intensity derivative ∂I(r)/∂z. Experimentally, the intensity is easy to obtain. The intensity derivative can be estimated by a finite difference between the two or more closely separated images (i.e. at different values of z). In prior art systems, this was done by acquiring an image stack with slight defocus, by translating the camera or the object manually or mechanically.
The TIE-based phase imaging technique has been increasingly investigated in recent years because of its unique advantages over interferometric techniques [8,9]: it is non-interferometric, works with partially coherent light sources, is computationally simple, does not need phase unwrapping, and does not require a complicated optical system. However, despite its evident merits and great improvements, TIE-phase imaging technology has still not gained as much attention or such widespread applications as interferometric techniques in quantitative phase microscopy. One important reason is that, as noted, the TIE typically requires a series of images captured at different focal depths that are usually realized by translating the camera or the object manually or mechanically. This not only complicates the image acquisition process, but also prolongs the measurement time, precluding real-time observation of a dynamic process.
Techniques have previously been proposed to avoid the need for mechanical motion in the stack acquisition. In [10], the image is captured by a volume holograph which produces a three-dimensional intensity image. In [11], the chromatic aberration inherent in any lens system is exploited, by comparing the intensity images associated with different respective light frequency ranges. In [12], the sample is caused to flow through the focal plane, so that images acquired at different times correspond to different object planes. However, it is still demanding to produce a simple, practical, non-mechanically controlled focus-variable system that can deliver performance near diffraction-limited imaging, for enabling high-speed TIE phase imaging for dynamics applications.